Solve ∫ (from 8.990 to 85.520) of 44137.115*10.5-0.32sinθ)*sinθcosθdθ

$$\int _ { 8.990 } ^ { 85.520 } 44137.115 \times 10.5 - 0.32 \sin \theta ) \times \sin \theta \cos \theta d \theta$$

Evaluate

$$-\frac{15306\sin(\theta )}{625}+35467040.814975$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \frac{21\times 44137.115}{2}-\frac{8\sin(\theta )}{25}\mathrm{d}x$$

Find the integral of \frac{21\times 44137.115}{2}-\frac{8\sin(\theta )}{25} using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\left(\frac{21\times 44137.115}{2}-\frac{8\sin(\theta )}{25}\right)x$$

Simplify.

$$\left(463439.7075-\frac{8\sin(\theta )}{25}\right)x$$

The definite integral is the antiderivative of the expression evaluated at the upper limit of integration minus the antiderivative evaluated at the lower limit of integration.

$$85.52\left(463439.7075-0.32\sin(\theta )\right)-8.99\left(463439.7075-0.32\sin(\theta )\right)$$

Simplify.

$$35467040.814975-\frac{15306\sin(\theta )}{625}$$

Differentiate w.r.t. θ

$$-\frac{15306\cos(\theta )}{625}$$